Inverse Of Y Equals X Squared Minus 10x A Step By Step Solution

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In the realm of mathematics, understanding inverse functions is crucial, especially when dealing with quadratic equations. This article delves into the process of finding the inverse of the quadratic function $y = x^2 - 10x$, providing a detailed, step-by-step solution and insightful explanations. Our primary focus will be on transforming the given equation to isolate x in terms of y, a procedure that unveils the inverse function. Through this exploration, we aim to enhance your understanding of inverse functions and their significance in mathematical problem-solving. The options presented are:

A. $y= \pm \sqrt{x-25}-5$ B. $y= \pm \sqrt{x-25}+5$ C. $y= \pm \sqrt{x+25}-5$ D. $y= \pm \sqrt{x+25}+5$

The Essence of Inverse Functions

Before we embark on the solution, it's essential to grasp the fundamental concept of inverse functions. An inverse function essentially reverses the operation of the original function. If a function f maps x to y, then its inverse, denoted as f⁻¹, maps y back to x. In simpler terms, the inverse function undoes what the original function does. This concept is vital in various mathematical applications, including solving equations, understanding transformations, and exploring symmetrical relationships.

However, not all functions have inverses. For a function to possess an inverse, it must be one-to-one, meaning each output corresponds to a unique input. Quadratic functions, like the one we're examining, are not one-to-one over their entire domain due to their parabolic shape. To find an inverse, we often restrict the domain of the original function to make it one-to-one. This restriction is a key step in the process, ensuring the inverse function is well-defined and mathematically sound.

Method 1: Step-by-Step Solution

Our journey to find the inverse of $y = x^2 - 10x$ involves several key steps. We'll employ the technique of completing the square, a method that transforms the quadratic expression into a more manageable form. This transformation allows us to isolate x effectively and ultimately derive the inverse function. Each step is meticulously explained, providing clarity and insight into the underlying mathematical principles.

  1. Rewriting the Equation: The first step is to rewrite the given equation, $y = x^2 - 10x$, in a form that facilitates completing the square. We aim to manipulate the equation to create a perfect square trinomial on one side. This process involves adding and subtracting a constant term, carefully chosen to maintain the equation's balance. The goal is to transform the quadratic expression into a squared term plus a constant, making it easier to isolate x. This initial transformation is crucial for the subsequent steps in finding the inverse function.

  2. Completing the Square: The heart of our solution lies in completing the square. To do this, we take half of the coefficient of the x term (-10), square it ((-5)² = 25), and add it to both sides of the equation. This process transforms the right side into a perfect square trinomial, which can be factored into the form (x - a)². This step is pivotal in isolating x, as it allows us to express the quadratic expression as a squared term, simplifying the equation and bringing us closer to finding the inverse.

    y+25=x2−10x+25y + 25 = x^2 - 10x + 25

    y+25=(x−5)2y + 25 = (x - 5)^2

  3. Isolating (x - 5)²: Now, we have the equation in a form where we can easily isolate the squared term. This isolation is a critical step in solving for x. By having (x - 5)² on one side of the equation, we can apply the square root operation to both sides, which is the next step in our journey to find the inverse function. The goal here is to progressively peel away the layers surrounding x, bringing us closer to expressing x in terms of y.

  4. Taking the Square Root: To eliminate the square, we take the square root of both sides of the equation. Remember, taking the square root introduces both positive and negative solutions, denoted by the ± symbol. This is a crucial point, as it reflects the nature of quadratic functions and their inverses. The ± symbol indicates that for a given value of y, there are potentially two corresponding values of x, highlighting the importance of considering both branches when dealing with inverse functions of quadratics.

    ±sqrty+25=x−5\pm \\sqrt{y + 25} = x - 5

  5. Solving for x: The final step in finding the inverse function is to isolate x. We achieve this by adding 5 to both sides of the equation. This gives us x expressed in terms of y, which is the essence of an inverse function. The resulting equation represents the inverse of the original function, mapping y back to x. This step completes the transformation, providing us with the inverse function in its explicit form.

    x=pmsqrty+25+5x = \\pm \\sqrt{y + 25} + 5

  6. Expressing the Inverse: To express the inverse function in standard notation, we swap x and y. This convention clarifies that we are dealing with the inverse, where the roles of input and output are reversed. The resulting equation, $y = \pm \sqrt{x + 25} + 5$, represents the inverse of the original function, clearly illustrating the mapping from y back to x.

    y=pmsqrtx+25+5y = \\pm \\sqrt{x + 25} + 5

  7. The correct answer is D: By method 1, we can clearly see that the answer is $y= \pm \sqrt{x+25}+5$, which is option D.

Conclusion

In conclusion, the inverse of the quadratic function $y = x^2 - 10x$ is $y = \pm \sqrt{x + 25} + 5$, corresponding to option D. This solution was derived through a meticulous process of completing the square, isolating the squared term, taking the square root, and solving for x. Understanding inverse functions is a cornerstone of mathematical proficiency. The ability to find inverses not only enhances problem-solving skills but also deepens the comprehension of function transformations and symmetrical relationships. This article has provided a comprehensive guide to finding the inverse of a quadratic function, equipping you with the knowledge and techniques to tackle similar challenges with confidence and precision. The process of finding inverse functions is not just a mathematical exercise; it's a journey into the heart of functional relationships, revealing the interconnectedness of mathematical concepts and their applications in various fields.